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МАТЕМАТИКА
Approximation of Continuous 2n-Periodic Piecewise Smooth Functions by Discrete Fourier Sums
G. G. Akniyev
Gasan G. Akniyev, https://orcid.org/0000-0001-8533-4277, Dagestan Scientific Center RAS, 45 M. Gadzhieva St., 367025 Makhachkala, Russia, [email protected]
Let N be a natural number greater than 1. Select N uniformly distributed points tk = 2nk/N + u (0 < k < N - 1), and denote by LnN(f) = LnN(f,x) (1 < n < N/2) the trigonometric polynomial of order n possessing the least quadratic deviation from f with respect to the system {tk}N—. Select m + 1 points —n = a0 < ai < ... < am-i < am = n, where m ^ 2, and denote O = {a,}m=0. Denote by C^ a class of 2n-periodic continuous functions f, where f is r-times differentiable on each segment A, = [a,, ai+i] and f(r) is absolutely continuous on A,. In the present article we consider the problem of approximation of functions f e C^ by the polynomials Ln,N(f,x). We show that instead of the estimate |f(x) — Ln,N(f,x)| < clnn/n, which follows from the well-known Lebesgue inequality, we found an exact order estimate |f(x) — Ln,N(f,x)| < c/n (x e R) which is uniform with respect to n (1 < n < N/2). Moreover, we found a local estimate |f (x) — Ln,N(f, x)| < c(e)/n2 (|x — a,| ^ e) which is also uniform with respect to n (1 < n < N/2). The proofs of these estimations are based on comparing of approximating properties of discrete and continuous finite Fourier series.
Keywords: function approximation, trigonometric polynomials, Fourier series.
Received: 22.05.2018 / Accepted: 28.11.2018
Published online: 28.02.2019
DOI: https://doi.org/10.18500/1816-9791 -2019-19-1 -4-15
INTRODUCTION
We begin by establishing some notations. Let O be a set of m +1 points {a,}m=0 (m > 2) such that —n = ao < < ai < ... < am-i < am = n. We denote by the class of 2n-periodic functions with r absolutely continuous derivatives on each interval (a, ,ai+i) and by CO the subclass of
all continuous functions in (here we say that a function f is absolutely continuous on an interval (a, b) if the function f is absolutely continuous on the segment [a, b], where f (x) = f (x) for x £ (a, b), f (a) = f (a + 0), and f(b) = f (b - 0)).
We denote by Ln,N(f, x) a trigonometric polynomial of order n possessing the least quadratic deviation from the function f at the points {tjj^-1, where tj = u + 2nj/N, n ^ N/2, N ^ 2, and u £ R. In other words, Ln,N(f, x) provides the minimum for the sum ij-1 |f(tj) — Tn(tj)|2 on the set of all trigonometric polynomials of order at most n. To read more about function approximation by trigonometric polynomials see [1-10].
Also, in this paper we denote by c or c(b1, b2,..., bk) some positive constants, which depend only on specified arguments (if any) and may vary from line to line, and by Sn(f, x) the n-th partial sum of the Fourier series of the function f. We also note that it is easy to show that the Fourier series of any f £ converges uniformly on R and the following representation is possible:
to
f (x) = 70 + ^^ (ak cos kx + bk sin kx), (1)
2 k=i
1 r 1 r
ak = -/ f (t)cos ktdt, bk = -/ f (t) sin ktdt. (2)
n J-n n J-n
where
11
— f (t) cos ktdt, bk = — n J-n n
The goal of this work is to estimate the value |f(x) — Ln,N(f, x)| for f £ . Note that the special case of this problem is considered in [11], where the value |f(x) — Ln,N(f,x)| is estimated for a 2n-periodic function f(x) = |x| (x £ [—n,n]). In this work, we generalize the results from [11] for any function f £ , as stated in the following theorem:
Theorem 1. For f £ the following inequalities hold:
|f(x) — Ln,N(f,x)| < f, x £ R, (3)
n
|f (x) — Ln,N(f, x)| < , x £ |x — a| ^ (4)
The order of these estimates cannot be improved. To prove this theorem we use a lemma from [12]:
Lemma 1 (Sharapudinov, [12]). If the Fourier series of f converges at the points tk = u + 2kn/N, then the representation
Ln,N (f, x) = Sn(f, x) + Rn,N (f, x), (5)
where
2 to }
Rn,N(f, x) = - Y^ Dn(x — t) cos^N(u — t)f (t)dt, (6)
holds true, where 2n < N and Dn (x) is the Dirichlet kernel:
2
Dn (x) = - + ^>s kx. (7)
k=1
This lemma considers only the case 2n < N .If 2n = N (when N is even) we can write (see [12])
Ln,2n (f,x) = Ln-l,2n (f,x) + (f )C0S n(x - u),
where
2n-1
ai2n) (/ ) = 2^E/ (tk )cos "(tk — «)•
k=0
To prove the inequalities (3) and (4) from Theorem 1 we use the formulas |/(x) — Ln,N(/, x)| ^ |/(x) — Sn(/,x)| + K,N(/, x)| , n < N/2,
(2n)
(8)
(9)
(10)
|/(x) — Ln,N(f,x)| < |/(x) — Sn-i(/,x)| + |Rn-i,N(/,x)| + |an (/)| , n = N/2, (11)
which immediately follow from (5) and (8).
The estimates for the values |/(x) — Sn(/,x)|, |Rn,N(/,x)|, and |an2n)(/)| are found in the following sections.
1 . THE ESTIMATE FOR |/(x) — Sn(/, x)|
To estimate the value |/(x) — Sn(/, x)| we need the following lemma.
Lemma 2. For / e C^ the following inequality holds:
f (t)hp (k(t + a))dt
<
cf) k2 '
where k G N, a G R, and
hp (x) =
cos x, p = 0,
sin x, p = 1.
Proof. Performing integration by parts two times we have
(12)
f(t)hp(k(t + a))dt = Ц^- I f (t)hi-p(k(t + a))dt
1 k2
—n m—1
Y. f (« - 0) - f' (« + 0)) hp(k(a, + a)) - f (t)hp(k(t + a))dt i=0 _
From this we can get the estimate
f (t)hp (k(t + a))dt
1
< k12
m—1
£|f' (« - 0) - f (a, + 0) + f (t)
i=0
dt
<
c(f) k2 ■
□
П
П
Lemma 3. For f G C2 the following inequalities hold:
|f (x) - Sn(f,x)| < f, x G R, (13)
n
|f (x) - Sn(f, x)| < fr, |x - O« | > £. (14)
Proof. Here we prove only (13) because the proof for inequality (14) can be found in [13, Theorem 2.1]. Using (1) and (2) we can write
to
f (x) — Sn (f, x) = ^^ (ak cos kx + sin kx).
k=n+1
Applying Lemma 2 to (2) we get |ak| ^ c(f)/k2 and |bk| ^ c(f)/k2, which gives us
to
|f(x) - S„(f,x)| ^ £ (|Ok| + |) < c(f)/n.
k=n+1
□
2 . THE ESTIMATE FOR |Rn,N(f, x)
From (6) and (7) follows (f,x) = Ri(f,x) + R^n(f,x), where
1 to }
(f, x) = f (t) cos mN(u - t)dt,
2 to n "
9 _ _ _ _ /*
R22(f, x) = - ^ f (t) cos k(x - t) cosmN(u - t)dt. (15)
^ ^=1 k=1-n
Obviously, |Rn,N(f,x)| < |Rn,N(f,x)| + |R2,n(f,x)|. The values |Rn,n(f,x)| and |Rn,N(f, x) | are estimated later in this section, but first we prove three auxiliary lemmas.
Lemma 4. For f £ C0'1 the following holds:
f (t)hp(k (t — x))hq (MN (t — u))dt =
—n m—1
( J2 E (f (Oi - 0) - f (Oi + 0)) hp(k(Oi - x))h1—q(mN(Oi - u))-
(mN)2 - k2 i=0
n
- /äff2 / f'(t)hp(k(t - x))h1—q(MN(t - u))dt+ («N) - k2 J
(mN r - '2 ' " P
—n
m—1
(_i)1+pk m—11
+ / ( a 12 , 2E (f (Oi - 0) - f (Oi + 0)) h1— p(k(Oi - x))hq(MN(Oi - u))-
(mn) - k21=0
n
(MN1)2-kkiJ f (t)h1—p(k(t - x))hq(mN(t - «))&. (16)
—n
Изв. Сарат. ун-та. Нов. сер. Сер. Математика. Механика. Информатика. 2019. Т. 19, вып. 1 Proof. Perform integration by parts:
f (t)hp(k(t - X))hq(MN(t - U))dt =
—n
m—1
(_1)q m—1
(f («i - 0) - f («, + 0)) hp(k(a, - x))h1—q(mN(a, - u))-
MN
i=0
- TNT / f' (t)hp (k(t - x))h1—q (mN (t - u))dt+
— П
7Г
+ ( m^V/(t)hi-p(k(t — x))hi-q(mN(t — u))dt. (17)
—n
Repeat integration by parts for the last integral in (17):
I /(t)hp(k(t — x))hq(mN(t — u))dt =
— П
m—1
(_1)q m—1
^ E (f (a, - 0) - f (a, + 0)) hp(k(a, - x))h1—q(mN(a, - u))-
i=0
(-1)q , ,
( ) ' f (t)hp(k(t - x))h1—q(mN(t - u))dt+
MN
— П
m—1
(_1)1+p k m—1
\ ;П2 E (f (a, - 0) - f (a, + 0)) h1— p(k(a, - x))hq(mN(a, - u))-(MN)
,=0
П
-^тНтг? f f'(t)h1—p(k(t - x))hq(mN(t - u))dt+ (mN ) J
(MN)
k2 (mn )
— П
f (t)hp(k(t - x))hq(mN(t - u))dt.
2p — П
By moving the last integral from the right side to the left and dividing both sides by
2 _
^NNT2 we get (16). □
Corollary 1. If f G CQ, then f (a, - 0) - f (a, + 0) = 0, so we can write (16) as
П П
J f (t)hp(k(t - x))hq(MN(t - u))dt = ¿ff -N J' f' (t)hp(k(t - x))h1—q(mN(t - u))dt-
— П —П
(-1)1+p k
- - ■■■"Я—p
J f (t)h1—p(k(t - x))hq(mN(t - u))dt.
(mN)2 - k2
— П
7Г
Lemma 5. The following estimate holds:
1
(kx)
k=1
<
Proof. The proof is obvious and follows from well-known formulas
£ sin(fex)=sin (:;n (f;, ± cos(kx)=cos (^xjsin (nx)
k=1
sin (2)
□
Lemma 6. Let a1, a2,..., an be a monotonous sequence (either increasing or decreasing) of n positive numbers. The following holds:
hp (kx)
k=1
<
2an + a
| sin x
Proof. After performing Abel transformation (summation by parts) we have:
n—1
^ akhp(kx) = an E hP(jx) - S (ak+1 - ak) E hP(jx)-
k=1
=1
k=1
=1
Using Lemma 5 and the fact that ^n=1 |ak+1 — ak| = En=1 ak+1 — ak| we can write
hp (kx)
k=1
< a
n—1
k=1
<
E hp(jx) j=1
' n—1
^«k+1 - ak
an +
+ y] |ak+1 - ak|
2an + a
hp(jx)
j=1
<
k=1
<
'sin x
□
Lemma 7. The following inequality holds: R N(f, x)| ^ c(f)/N2. Proof. Using Lemma 2 we have
Rn(f,x)| < ^
^=1
f (t) cos mN(u - t)dt
< f v 1 < f
< N2 ^ M2 < N2
U = 1
Lemma 8. The following estimates hold:
K,n(f,x)| < ,
x e R,
R2(f,x)| <
f)
N2
, |x - O; | ^ e.
□
(18) (19)
2
1
2
n
Proof. Rewrite (15) using (12):
^ Ж n П
R(f,x) = -££ f(t)ho(k(t - x))ho(MN(t - u))dt.
П
"=1 fc=1_
Using Corollary 1 we rewrite the above formula as follows:
СЛ Ж 1 n -1 /»
R2n,N(f,x) = E " E-f (t)ho(k(t - x))h1 (mN(t - u))dt+
"=1 M k=1 1 - (JN) —П
2 1 k 1 /* +r^E^E N—/' (t)hi (k(t — x))ho (mn (t — =
m k=iN1 — (¿N) ——n
= r^N (/,x) + Rjv (/,x).
For brevity we only consider here estimation of |R22,',n(/, x) | because IR'N(/, x) | can be estimated in almost the same way. Obviously, /' e C^'1, so we can apply Lemma 4 to R" (/, x):
- ж 1 n 1 m—1
(f x) = ^Е зЕ ( - ^E (f' (a® - 0) - f' (a + °))
k=1 (1 - (») '=°
xho(k(a, - x))ho(mN(a, - u)
x
2 1 1 С ^ E E T-^ f''(t)ho(k(t - x))ho(MN(i - u))dt+
k=1 (! - O) )
—2 A 1 -A k m—1
+^ E M3 E T : V E f (a, - 0) - f (a, + 0)) x
k=1 (1 - O)) ,:=o
xh1 (k(a, - x))h1 (mN(a, - u)) +
2 ж 1 n k П
E 7! E T-^ / f''(t)h1 (k(t - x))h1(MN(t - u))dt
nN( l , -2\
" k=111 - O))
= r2,N (f,x) + RtN (f,x) + r2,N (f,x) + r2;,N4 (f,x).
Begin with R2'1v (f,x). Applying Lemma 4 we get
q ж n m—1
R^N (f, x) = ^E ^E t / -^E (f "(a. - 0) - /''(a, + 0)) x
k=1 (1 - O)) !=°
xho(k(a, - x))h1 (mN(a, - u)) + 10 Научный отдел
П
G. G. Akniyev. Approximation of Continuous -Periodic Piecewise Smooth Functions
—2 ^ ^_1
nN3 ^ -3 f^
¿ = 1 ^ k = 1
f''' (t)ho (k(t — x))h1 (-N (t — u))dt+
1 - ( A i i -n
¿N
k
m-1
_2 to i n
+ nN4 ^ J4 k=1 ( u\ 2 . . 0
¿=1 r k=1 M _ / \ 1 i=0
1 Un
sE (f''(ai — 0) — f'' (a, + 0)) x
xh1 (k(a, — x))h0(-N (a, — u))+
2 to i n ttTV4
k
nN4 ^ -4 ^ / / x2\
k=1 (i — (¿v))
f''' (t)h1(k (t — x))ho (-N (t — u))dt.
From this we can get the estimate
to n
c1
m- 1
KN2(f,x)| <
N3 ^ J3 ^ / , ,2 ¿= - k=1 11 — ^
^|f'' (a, — 0) — f'' (a, + 0)
i=0
+
C to 1 n
+ N3 ^ -3 ^ ¿=1 ^ k=1
C to 1 n
+ N3 ^ J4^ ¿=1 ^ k=1
k/N
1 — '¿N
m- 1
fm (t)
dt+
1 — ' ¿N
i=0
T^Ef'' (a' — 0) —
+
TO 1 n
3 E-^E
k/N
N ¿= - " (1 — (*)=)
f'" (t)
dt <
Cf) N2 '
In the same way we can get R^N4(f, x) | ^ c(f )/N2. Now we consider R^N1 (f,x and | Rn,vv3(f, x) |. We will estimate here only R^v1 (f, x) | because the other one can be estimated in the similar way. After a simple transformation we have
2 to 1 m-1
R^'N1 (f, x) = -^E " E f' (a, — 0) — f' (a, + 0)) M/'N (a, — u)) E
n ¿=1 - ,=0
k=1
hp (k(a, — x))
'1 — i ¿v
From this we have the uniform estimate for x R:
to m-1
|
R'N1 (f,x)| < ^E -2 E |f' (a, — 0) — f' (a, +
¿=1 - ,=0
E
k=1
h0 (k(a, — x))
1 — ' ¿N
<
nc(f) N2
n
3
n
3
1
n
1
n
3
2
Maтeмaтnкa
11
Using Lemma 6 and assuming = --1—^ we have
E
k=1
hp(k (a« — x))
<
sin
i-( TN )
2
V 1 — W
T7 +
1
1—
<
I sin
Now we can write
KN1 (f,x)| <
¿=1 ^ i=0
In the similar way we can get
nc(f)
c ^ 1 m-1 If (a« — 0) — f (a« + 0)| < c(f,e)
I sin ^
N2
|x — a«| ^ e.
c(f,e)
(f x)I < ^, x € R and (f,x)I < ^Z, |x — a«| > e.
Finally, for R.'N (f, x) we can write
N2
(f x)| < EKN(f, x) I < if, x € R, IR22',N(f,x)| < f-, |x — a«| ^ e
«=1 N N
Using the same approach we can show that the value IR'jv(f, x) | has the same estimate as IR'iv(f, x)|, which leads us to (18) and (19). □
From the previous lemmas and inequality (f,x)| ^ R(f,x)| + (f,x follow estimates for |Rn>N(f, x) |:
|Rn,N(f,x)| < nf, x € R, |R„,n(f, x)| < , |x — a«| > e.
(20) (21)
3 . THE ESTIMATE FOR
ai2n) (f)
Lemma 9. For the value ai2n) (f) where f
ai2n) (f) < c(f )/N2 holds.
Proof. For each f € Cg the sum S = £k=—1 (ffe) — f(tfe+1)) = 0. We can
represent the above sum as S = S1 + S2, where S1 = ^П=0 (f (t2k) — f (t2k+1)) and
S2 = £n-01 (f (W) - f (t2k+2)). We can see that Si = -S2 and |Si| = |Si - S21 /2. From the above formulas we can write the equation for S1 — S2:
n=1
n=1
S1 — S. = £ (f (t2k) — 2f (i.fc+1) + f (t.k+2)) = £ Д2f (t2k) .
(22)
k=0
k=0
Denote by G the set of numbers Um=0 {k : 0 < k < n, |t2k+1 — a«| < 2^} and
G = {k}J!=0 \ G. Rewrite (22) by dividing it into two sums:
S1 — s. = E д2 f ) + E Д2 f ((2* )•
fceG
fceG
1
c
2
2
a =x
a — x
2
2
2
For every k G G we have |t2k+1 — a«| > 2n/N (0 ^ i ^ m) so the points t2k, t2k+i, t2k+2 are inside some interval (ai5ai+1) and the function f G C^ has an absolutely continuous derivative f'' on (a^, ai+1), therefore we can write |A2f(t2k)| ^ 2maxxG[_n,n] |f''(x)|. For k G G we can write |A2f(t2k)| ^ c(f)/N, also note that |G| ^ 2m. Therefore, we have
|Si| = ^^ < f. (23)
From (9)
2n_1 n_1
ai,2n)(f) = f (tk) cosnk = N ^ (f (i2fc) — f (i2fc+1)) = NS1. (24)
k—0 k—0
From (23) and (24) follows a(N)(f) < c(f )/N2. □
4. THE PROOF OF THEOREM 1
The proof of Theorem 1 consists of two parts: first we prove that the inequalities (3) and (4) of the theorem hold, then we prove that these estimates cannot be improved for all f G C2.
From the inequalities (10), (11), the estimates (13), (14), (20), (21) and Lemma 9 we can easily get (3) and (4). To prove that the order of these estimates cannot be improved we consider the aforementioned 2n-periodic function f (x) = |x|, x G [—n,n]. Obviously, f G . Consider only the case when n < N/2. From (5) follows the inequality |f(x) — Ln,N(f,x)| ^ |f(x) — Sn(f,x)| — |Rn,N(f,x)|. From (20) follows |Rn,N(f, x)| ^ c(f)/N. Therefore, for every e > 0 we can find a natural number N0, such that for every N > N0 follows |Rn,N(f,x)| < e. Let N0(n) be a natural number such that for every N > N0 (n)
max |Rn,N(f,x)| ^ 1 max |f (x) — Sn(f,x)| ,
xGE 2 xGE
N>N0 (n)
where E c R. So, we can write
max |f(x) — Ln,N(f, x)| ^ 1 max |f (x) — Sn(f, x)|. (25)
XGE 2 xGE
N>No(n)
Lemma 10. 7he following inequalities hold:
max |f (x) — Sn(f,x)| ^ c(f)/n, x G R,
xGR
max |f (x) — Sn(f, x)| ^ c(f, e)/n2, |nk — x| ^ e.
Proof. From [14, p. 443] we have the following representation:
, , , n 4 ^ cos(2k — l)x r ,
f(x) = |x| = 2 — n £ (2k — I)* ' x G [—n'n]'
k—1
From the previous equation we can get f (x) - (f,x) = -4 ^м--)^ • For
x = 0 we have
4 1
R(f,0)1 = 4 £ (2k—1)2 > c/n
k=n+1 V J
Now we consider the case when x = n/4 and n + 1 = 41, 1 G N. It is easy to show that
R f q = _2^2 v ( 1 i 1___1___—^ =
ПУ "4) п (8k - 1)2 (8k + 1)2 (8k + 3)2 (8k + 5)V
8k + 1 8k + 3 \
+
n y (8k - 1)2 (8k + 3)2 (8k + 1)2 (8k + 5)2 / '
Hence we have |Rn f, | ^ c/n2. □
From (25) and the above lemma follows
max f (x) - (f, x) | ^ C, x G R,
N>N0 (n)
max |f(x) - Ln,N(f,x)| ^ ^, |x - nk| ^ 6.
N>N0 (n)
Theorem 1 is proved. References
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Cite this article as:
Akniyev G. G. Approximation of Continuous 2n-Periodic Piecewise Smooth Functions by Discrete Fourier Sums. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2019, vol. 19, iss. 1, pp. 4-15. DOI: https://doi.org/10.18500/1816-9791-2019-19-1-4-15
УДК 517.521.2
Приближение непрерывных 2п-периодических кусочно-гладких функций
дискретными суммами Фурье
Г. Г. Акниев
Акниев Гасан Гарунович, младший научный сотрудник, Дагестанский научный центр РАН, 367025, Россия, Махачкала, ул. М. Гаджиева, д. 45, [email protected]
Пусть N > 2 — некоторое натуральное число. Выберем на вещественной оси N равномерно расположенных точек = 2nk/N + u (0 < k < N - 1). Обозначим через (f) = (f, x) (1 < n < N/2) тригонометрический полином порядка n, обладающий наименьшим квадратичным отклонением от f относительно системы ■ Выберем m + 1 точку
-п = a0 < a1 < ... < am-1 < am = п, где m > 2, и обозначим О = {a«}™0. Через C^ обозначим класс 2п-периодических непрерывных функций f, r-раз дифференцируемых на каждом сегменте Д« = [a, ai+1], причем производная f(r) на каждом Д« абсолютно непрерывна. В данной работе рассмотрена задача приближения функций f е полиномами (f, x). Показано, что вместо оценки |f (x) - (f, x)| < c ln n/n, которая следует из известного неравенства Лебега, найдена точная по порядку оценка |f (x) - (f, x)| < c/n (x е R), которая равномерна относительно n (1 < n < N/2). Кроме того, найдена локальная оценка |f (x) - (f, x)| < c(e)/n2 (|x - a«| > e), которая также равномерна относительно n (1 < n < N/2). Доказательства этих оценок основаны на сравнении дискретных и непрерывных конечных сумм ряда Фурье.
Ключевые слова: приближение функций, тригонометрические полиномы, ряд Фурье. Поступила в редакцию: 22.05.2018 / Принята: 28.11.2018 / Опубликована онлайн: 28.02.2019
Образец для цитирования:
Akniyev G. G. Approximation of Continuous 2n-Periodic Piecewise Smooth Functions by Discrete Fourier Sums [Акниев Г. Г. Приближение непрерывных 2п-периодических кусочно-гладких функций дискретными суммами Фурье] // Изв. Сарат. ун-та. Нов. сер. Сер. Математика. Механика. Информатика. 2019. Т. 19, вып. 1. С. 4-15. DOI: https://doi.org/l0.18500/1816-9791-2019-19-1-4-15
